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The concept of self‐organization proposed by Von Foerster and developed recently by I. Prigogine and the “synergetics” of H. Haken state a framework that promises to be fruitful in the construction of theories synthesizing the microscale and the macroscale points of view in complex systems. This concept may be especially interesting in human sciences, like sociology, where the duality microsociology/ macrosociology remains. Tries to identify explicitly the main features defining the self‐organization of microelements that produce a macroscopic system, and applies the concept to phenomena of social evolution, suggesting a formulation of the micro/macro relationship in social sciences in terms of the probabilistic field theory.

A general method is proposed to approximate the analytical solution of any time‐dependent partial differential equation with boundary conditions defined on the four sides of a rectangle. To ensure that the approximant satisfies the boundary conditions problem the differential operator is modified with one additional term which takes into account the effect of boundary conditions. Then the new problem can be directly integrated in the same way as an ordinary differential equation. In this work Adomian's decomposition method with analytic extension is used to obtain the first‐order approximant to the solution of a test case. The result is an analytic approximation to the solution which is compatible with both the exact boundary conditions and the accuracy imposed in the whole domain. The solution obtained is compared with the analytic approximation obtained with a Tau‐Legendre spectral method.

Adomian's method is completed to obtain the analytic solution of any partial differential equation with boundary conditions defined on the four sides of a rectangle. Adomian's decomposition method is first used to obtain the N‐order approximant to the one direction partial solution that satisfies the boundary conditions on that direction. Then the functions obtained are variationally modified on the four sides to make them compatible with a given experimental error. The product of these transformations is an analytic approximation to the solution which is compatible with both the weak norm imposed on the boundaries and the accuracy imposed in the whole domain.